Introduction to Graphing: Solutions

Here are solutions to the questions on the worksheet accompanying the section Introduction to Graphing.  

• What is the name of the function that is initially being graphed?  

  Answer

  Looking at the list of functions on the left, we see that "abs" is selected.  This looks like the absolute value function.

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• What is the "plot-window" (i.e., the limits on the horizontal and vertical axes) for this plot?

  Answer

  Looking at the control boxes on the right, or by estimating from the graph itself, we see that the plot only shows the values between -5 and 5 on both the horizontal and vertical axes..

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• Describe precisely what happens to limits on the "plot-window" when you hit the "Zoom In", "Zoom Out", and "Restore" buttons?  

  Answer

  In this case, "Zoom In" divides the limits by 2, while "Zoom Out" multiplies the limits by 2.  That is because the graph happens to be centered on the origin, (0, 0).  In general, the center point of graph is kept the same, while the size of the horizontal and vertical range is divided or doubled.  In any case, the "Restore" button returns the limits to their original settings.

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• What are the eight commands listed in the top menu?  

  Answer

  Clicking on the menu reveals the commands:

  • Main Screen
  • Multigraph Utility
  • Animate Utility
  • Parametric Curves Utility
  • Derivatives Utility
  • Riemann Sums Utility
  • Integral Curves Utility
  • Graph 3D Utility

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• What value does the program give for the absolute value of -3?  

  Answer

  The program displays the equation abs(-3) = 3.  That is, the absolute value function takes an input of -3 to an output of 3.  We normally write this as |-3| = 3.  This corresponds to the point on the graph (-3, 3).

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• Give a rule, in words, that describes the floor function. 

  Answer

  Looking at the results:

  x	floor(x)
  -2.5 3 4.12	-3 3 4

  It rounds down to the next lower integer.

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• Describe all the changes you observe in the "Main Screen" window. 

  Answer

  In the list of functions on the left, we can see that our new function, f, now appears.  The plot of f also appears as a fairly steep parabola opening upwards.

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• From the graph, estimate the domain and range of g.

  Answer

  The graph seems to go off to positive "infinity" vertically, as it goes to negative infinity horizontally.  It also appears that the function decreases in value, as we move from left to right, ending at the point (3, -5).  This means that the domain must be (-¥, 3] (i.e., all x values less than or equal to 3) and the range is [-5, ¥) (i.e., all y values greater than or equal to -5).

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• Use the "x =" box to calculate h(-2), h(-1), h(0), h(1), h(2), and h(3).  Note: This is very similar to what we did by hand in a previous Exercise; in this way, you will be able to use XFunctions to create your own Exercises and check your own work.

  Answer

  Entering each value, in turn, gives the following table of values:

  x	h(x)
  -2 -1 0 1 2 3	12 -3 -1 1 undefined undefined

  which corresponds to the formula:

  since this rule does not include values for x = 2 or x = 3.  Notice how it gives a value for x = -2, even though the y-value is too large for it to appear inside the plot window.

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• Use the "x =" box to calculate k(-2), k(-1), k(-0.5), k(1.5), and k(3).  Notice how it actually extends the domain of the function to include values that are not in our original table, but not over the entire real line.

  Answer

  Entering each value, in turn, gives the following table of values:

  x	k(x)
  -2 -1 -0.5 1.5 3	1 -2 -1.5 2 undefined

  Notice how, by specifying the function to be "piecewise linear", it connected-the-dots to extend the domain to all values between -2 and 2, so we obtain values for points that were not in the original table.  However, it does not gives a value for x = 3, since that is outside the range of values in the table.

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• Use the "x =" box to calculate q(-2), q(-1), q(-0.5), q(1.5), and q(3).  Compare these values with those of k.

  Answer

  Entering each value, in turn, gives the following table of approximate values:

  x	q(x)
  -2 -1 -0.5 1.5 3	1 -2 -2.123 3.931 undefined

  At x = -2, -1, 0, 1, 2, we obtain the same values as k, since that is how we constructed the graph.  However, this time the function is extended in very complex way to all other values between -2 and 2.  Again, XFunctions will not give a value for x = 3, since it is outside the specified domain of the function (i.e., the established horizontal limits of the graph).

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• Compare the graphs of the two functions algebraically (i.e., the two formulas) and geometrically.  Try to relate the two comparisons.

  Answer

  In mathematical notation, the functions are  y = ëxû and y = 2ëxû, that is, the formula for the second function is simply 2 times that of the first.  This means that, for any given x-value, the y-value of the second is always twice that of the first.  The second graph appears to be "steeper" than the first, because all of the "steps" are twice (i.e., 2 times) as high.

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• Compare the graphs of y = ëxû and y = -ëxû algebraically and geometrically (remember this is just mathematical notation for the two functions you have just plotted).  Try to relate the two comparisons.

  Answer

  This time, the formula for the second function is simply negative that of the first.  This means that, for any given x-value, the y-value of the second is always negative that of the first.  The second graph appears to be the same as the first, but flipped over.  We know that multiplying by -1 flips over the number line; since we are multiplying y-values, we can imagine this graph as the result of flipping the vertical axis.

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Go to Arithmetic and Graphing.

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Send questions or comments to jwicks@northpark.edu

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